Department of Economics
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WORKING PAPERS IN ECONOMICS
No 605
Genuine Saving and Conspicuous Consumption
Thomas Aronsson and Olof Johansson-Stenman
November 2014
ISSN 1403-2473 (print)
ISSN 1403-2465 (online)
Genuine Saving and Conspicuous Consumption
**Thomas Aronsson* and Olof Johansson-Stenman+
November 2014
Abstract
Much evidence suggests that people are concerned with their relative consumption, i.e., their consumption in relation to the consumption of others. Yet, the social costs of conspicuous consumption have so far played little (or no) role in savings-based indicators of sustainable development. The present paper examines the implications of such behavior for measures of sustainable development by deriving analogues to genuine saving when people are concerned with their relative consumption. Unless the resource allocation is a social optimum, an indicator of positional externalities must be added to genuine saving to arrive at the proper measure of intertemporal welfare change. A numerical example based on U.S. and Swedish data suggests that conventional measures of genuine saving (which do not reflect conspicuous consumption) are likely to largely overestimate this welfare change. We also show how relative consumption concerns affect the way public investment ought to be reflected in genuine saving.
JEL classification: D03, D60, D62, E21, H21, I31, Q56.
Keywords: Welfare change, investment, saving, relative consumption.
** The authors would like to thank Sofia Lundberg and Karl-Gustaf Löfgren for helpful comments and suggestions, and Catia Cialani for collecting and organizing data. Research grants from the Swedish Research Council (ref 421-2010-1420) are gratefully acknowledged.
*Address: Department of Economics, Umeå School of Business and Economics, Umeå University, SE – 901 87 Umeå, Sweden. E-mail: Thomas.Aronsson@econ.umu.se
+ Address: Department of Economics, School of Business, Economics and Law, University of Gothenburg, SE – 405 30 Gothenburg, Sweden. E-mail: Olof.Johansson@economics.gu.se
1. Introduction
The concept of genuine saving has gained much attention in literature on welfare measurement in dynamic economies. Genuine saving is an indicator of comprehensive net investment in the sense of summarizing the value of all capital formation undertaken by society over a time period. Earlier research shows that the genuine saving constitutes an exact measure of welfare change over a short time interval if the resource allocation is first best.1 Furthermore, in the aftermath of the World Commission on Environment and Development, genuine saving has also become an indicator of sustainable development. The World Commission defines the development to be sustainable if it meets “the needs of the present without compromising the ability of future generations to meet their own needs” (Our Common Future, 1987, page 54). One possible interpretation (discussed by, e.g., Arrow et al., 2003) is that sustainable development requires welfare to be non-declining, meaning that genuine saving becomes an exact indicator of sustainable development over a short time interval. Another is that the instantaneous utility must not exceed its maximum sustainable level, on the basis of which Pezzey (2004) shows that non-positive genuine saving constitutes an indicator of unsustainable development (although positive genuine saving does not necessarily imply that development is sustainable). In either case, genuine saving gives information of clear practical relevance for economic welfare.2
Yet, the literature dealing with genuine saving has so far focused on traditional neoclassical textbook models, where people derive utility solely from their own absolute consumption of goods and services (broadly defined). As such, it neglects the possibility discussed in the behavioral economics literature that people also enjoy consuming more, and dislike consuming less than others – an idea that appeared rather obvious to many leading economists of the past such as Adam Smith, John Stuart Mill, Karl Marx, Alfred Marshall,
1 The seminal contributions are Pearce and Atkinson (1993) and Hamilton (1994, 1996). See also Hamilton (2010) for a recent overview of the literature and van der Ploeg (2010) for a political economy analysis of genuine saving. Some other extensions beyond the standard model are found in Aronsson, Cialani, and Löfgren (2012) and Li and Löfgren (2012). The former derives a second-best analogue to genuine saving in a representative agent model with distortionary taxes and public debt, and the latter addresses genuine saving when growth is stochastic. See also the interesting, recent empirical application by Greasley et al. (2014), who test the welfare significance of genuine saving based on historical data from Britain.
2 This is further emphasized by the attention paid to genuine saving by the World Bank, which regularly publishes estimates of genuine saving for a large number of countries.
Thorstein Veblen and Arthur Pigou, before it became unfashionable in the beginning of the 20th century.
The purpose of the present paper is to examine how relative consumption concerns, through a preference for “keeping-up-with-the-Joneses,” affect the principles for measuring welfare change. It will then present a correspondingly adjusted measure of genuine saving.
Arguably, such a study is relevant for several reasons. First, there is now a large body of empirical evidence showing that people are concerned with their relative consumption, i.e., their consumption compared with that of referent others (and not just their absolute consumption as in standard economic models).3 Several studies find that increased relative consumption has a strong effect on individual well-being: questionnaire-experimental research often concludes that 30-50 percent of an individual’s utility gain from increased consumption may actually be due to increased relative consumption (e.g., Alpizar et al., 2005;
Solnick and Hemenway, 2005; Carlsson et al., 2007). Similarly, happiness-based studies typically find that a large (or even dominating) share of consumption-induced well-being in industrialized countries is due to relative effects (e.g., Luttmer, 2005; Easterlin, 2001;
Easterlin et al., 2010). In turn, this may distort the incentives underlying capital formation.
Second, based on the estimates referred to above, wasteful conspicuous consumption is likely to result in significant welfare costs, which – if not properly internalized – change the principles for calculating welfare change-equivalent measures of saving. Indeed, we show that the more positional people are on average, the more conventional models of genuine saving (where people are assumed to have standard utility functions) will overestimate the true welfare change. Third, recent literature shows that optimal policy rules for public expenditure are modified in response to relative consumption concerns,4 suggesting that such concerns may also affect the value of public investment in the context of genuine saving. This will be further discussed below.
3 See, e.g., Easterlin (2001), Johansson-Stenman et al. (2002), Blanchflower and Oswald (2004), Ferrer-i- Carbonell (2005), Luttmer (2005), Solnick and Hemenway (2005), Carlsson et al. (2007), Clark and Senik (2010), and Corazzini, Esposito, and Majorano (2012). See also Fliessbach et al. (2007) and Dohmen et al.
(2011) for evidence based on brain science and Rayo and Becker (2007) for an evolutionary approach.
4 See Ng (1987), Brekke and Howarth (2002), Aronsson and Johansson-Stenman (2008, forthcoming), and Wendner and Goulder (2008), who analyze different aspects of public good provision in economies where people are concerned with their relative consumption.
We develop a dynamic general equilibrium model where each consumer derives utility from his/her own consumption and use of leisure, respectively, and from his/her relative consumption compared with a reference consumption level (reflecting other people’s consumption). In the benchmark model, the relative consumption comparisons are of the keeping-up-with-the-Joneses type, meaning that each individual compares his/her current consumption with the current consumption of referent others, which is the case that best corresponds to the empirical evidence discussed above. However, we will also – although briefly – touch upon catching-up-with-the-Joneses comparisons, where the reference measure refers to other people’s past consumption, and argue that the associated externalities affect the welfare change measure in the same general way as the externalities following from keeping- up-with-the-Joneses types of comparisons.
Our main contribution is that we show how positional concerns influence the way welfare- change equivalent savings ought to be measured. We distinguish between a social optimum where all externalities are internalized, and unregulated market economies without externality correction. We also distinguish between first-best and second-best social optima by extending the benchmark model to allow for asymmetric information between the consumers and the social planner (or government). Furthermore, by using insights developed in the literature on tax and other policy responses to relative consumption concerns, we are also able to relate genuine saving to empirical measures of “degrees of positionality,” i.e., the extent to which relative consumption is important for individual well-being.
The paper closest in spirit to ours is Aronsson and Löfgren (2008). They consider the problem of calculating an analogue to Weitzman’s (1976) welfare-equivalent net national product in an economy where the consumers are characterized by habit formation. Their results show that if the habits are fully internalized through consumer choices, habit formation does not change the basic principles for measuring welfare (except that the individual’s own past consumption affects his/her current instantaneous utility). However, with external habit formation, i.e., if the habits partly reflect other people’s past consumption, the present value of this marginal externality affects the welfare measure through an addition to the comprehensive net national product. Our study differs from Aronsson and Löfgren (2008) in at least four distinct ways: we (i) consider measures of genuine saving (or analogues thereof) instead of net national product measures, (ii) focus attention on the empirically well- established keeping-up-with-Joneses type of comparison, (iii) allow for redistributive aspects
by considering a case where agents differ in productivity, and (iv) introduce public investments into the study of welfare change-equivalent savings.
The paper is outlined as follows. In Section 2, we present the benchmark model where each individual derives utility from consuming more than other people. We also present useful indicators of the extent to which relative consumption matters for individual well- being. Following earlier literature on genuine saving, we assume in the benchmark model that individuals are identical. In Section 3, we use the benchmark model to analyze economy-wide measures of welfare change. Sections 4 and 5 present two extensions by addressing catching- up-with-the-Joneses comparisons and public investments, respectively. Section 6 examines a more general model with two ability types that differ in productivity, where productivity is private information not observable to the social planner. Such a model allows us to extend the welfare analysis to a second-best model that includes both redistribution and externality correction subject to an incentive constraint. Section 7 presents a numerical example based on data for Sweden and the U.S. Section 8 concludes the paper.
2. The Benchmark Model and Equilibrium
Consider an economy with a constant population comprising identical individuals, whose number is normalized to one.5 The assumption of identical individuals is made for purposes of simplification; all qualitative results that we derive for this representative agent model would carry over in a natural way to a framework with heterogeneous consumers, as long as the redistribution policy can be implemented through lump-sum taxation.
Let c denote private consumption and z leisure. In a way similar to the models analyzed in Aronsson and Johansson-Stenman (2008, 2010), the instantaneous utility function faced by the representative individual takes the form
( , , ) ( , , )
t t t t t t t
U =u c z ∆ =u c z c . (1)
In equation (1), the variable ∆ = − denotes the relative consumption of the individual t ct ct and is defined as the difference between the individual’s own consumption and a reference
5 To be able to focus on the implications of relative consumption concerns in a simple way, we abstract from population growth. Genuine saving under population growth is addressed by Pezzey (2004). See also Asheim (2004).
measure, c .t 6 Each individual behaves atomistically and treats c as exogenous. The t assumption that the individual’s relative consumption reflects a difference comparison is made for technical convenience: all qualitative results derived below will also follow – yet with slightly more complex mathematical expressions – if the difference comparison is replaced with a ratio comparison (in which the relative consumption would become ct /ct).
The function u( )⋅ defines the instantaneous utility in terms of the individual’s absolute consumption and use of leisure, respectively, as well as in terms of the individual’s relative consumption compared with others, while the function u( )⋅ is a reduced form used in some of the calculations below. We assume that the function ( )u ⋅ is increasing in each argument and strictly concave, implying that u( )⋅ is increasing in its first two arguments and decreasing in the third. To be more specific, following equation (1) the relationships between the functions u ⋅ and ( )( ) u ⋅ are uc=uc+u∆, u = and z uz uc = − , where subscripts denote partial u∆ derivatives.
We follow Johansson-Stenman et al. (2002) and define the “degree of positionality” as a measure of the extent to which relative consumption matters for individual utility compared with absolute consumption. To be more specific, the degree of positionality measures the share of the overall instantaneous utility gain from increased consumption that is due to increased relative consumption. By using the function u( )⋅ , which distinguishes between absolute and relative consumption, the degree of positionality at time t can be written as
( , , )
(0,1) ( , , ) ( , , )
t t t
t
c t t t t t t
u c z
u c z u c z
α ∆
∆
= ∆ ∈
∆ + ∆ for all t. (2)
Therefore, 1− measures the degree of non-positionality, i.e. the extent to which the αt instantaneous utility gain of increased consumption is due to increased absolute consumption – an entity that is always set to unity in standard economic models. As indicated in the introduction, substantial empirical evidence suggests that the degree of positionality on average is in the interval 0.3-0.8 for income (which is interpretable as a proxy for overall consumption) in industrialized countries, while it may be even higher for certain visible goods such as houses and cars.
6 Note also that leisure is assumed to be completely non-positional. This is of course questionable, yet the limited empirical evidence available suggests that private consumption or income is much more positional than leisure (Solnick and Hemenway, 2005; Carlsson et al., 2007).
The objective faced by each consumer is the present value of future utility. If expressed in terms of the function u( )⋅ , the intertemporal objective function can then be written as (if measured at time 0)
0 0
( , , )
t t
t t t t
U e θ dt u c z c e θdt
∞ ∞
− = −
∫ ∫
, (3)where θ is the utility discount rate.
The usefulness of genuine saving as a measure of welfare change does not in any way depend on the number of capital stocks in the economy. Therefore, to simplify the model as much as possible we refrain from considering other types of capital than physical capital. In Section 5, we extend the model by incorporating public investment to show how the treatment of such investment in genuine saving reflects the policy rule for provision of the public good.
Let l denote the hours of work, defined by a time endowment, l , less the time spent on leisure, and k denote the physical capital stock. Output is produced by a constant returns to scale technology with production function ( , )f l k , which is such that fl > , 0 fk > , 0 fll < 0 and fkk ≤ .0 7 We suppress depreciation of physical capital, as it is of no concern in our context. This means that f( )⋅ is interpretable as net output (or that the depreciation rate is zero). The net investment at time t is then written in terms of the resource constraint as
( , )
t t t t
k = f l k −c, (4)
where the initial (time zero) capital stock, k , is fixed and lim0 t→∞kt ≥ . 0
The social decision problem is to choose c and t l for all t to maximize the present value t of future utility given in equation (3), subject to the resource constraint in equation (4), the initial capital stock, and the terminal condition. The present value Hamiltonian of this problem is given by (if written in terms of the utility formulation u( )⋅ in equation (1))
( , , ) [ ( , ) ]
p t p
t t t t t t t t
H =u c z c e−θ +l f l k −c , (5) where l denotes the costate variable attached to the capital stock and superscript p denotes present value. Note also that we are considering a representative-agent economy, where
t t
c = . In addition to equation (4) and the initial condition, the social first-order conditions c include
7 Note that the possibility of fkk =0 means that the model is consistent with an A-K structure, such that the economy grows at a constant rate in the steady state.
[ ( , , )uc c z ct t t +uc( , , )]c z ct t t e−θt =ltp (6a) ( , , ) t p ( , )
z c z c et t t θ t f l kl t t
u − =l (6b)
( , )
p
p t p
t t k t t
t
H f l k
l ∂ k l
= − = −
∂ , (6c)
where subscripts attached to the instantaneous utility and production functions denote partial derivatives. For further use, we also assume that the transversality conditions
limt→∞ltp ≥0 (=0 if limt→∞kt > ) 0 (6d)
limt→∞Htp =0 (6e)
are fulfilled.8 Note that the left-hand side of equation (6a) reflects the social marginal utility of consumption, u uc+ c = , since the social planner recognizes that relative consumption is uc social waste.
In an unregulated economy where the consumption externality is uninternalized, the social first-order condition for private consumption given by equation (6a) is not satisfied. Instead of introducing the decision problems faced by consumers and firms in the unregulated economy and then characterizing the general equilibrium, we just note that the outcome of such an economy would be equivalent to the special case of the model set out above where the social planner (erroneously) treats c as exogenous for all t. The first-order condition for t consumption would then change to
( , , ) t p
c c z c et t t θ t
u − =l , (7)
whereas the first-order condition for work hours and equation of motion for the costate variable remain as in equations (6b) and (6c), respectively.
3. Measuring Welfare Change in the Benchmark Model
This section presents measures of welfare change based on the benchmark model set out above. We begin by considering welfare change measures under first-best conditions in Subsection 3.1 and continue with the unregulated economy in Subsection 3.2. Some extensions of the benchmark model are discussed in Sections 4 and 5.
3.1 First-Best Resource Allocation
8 For a more rigorous analysis of transversality conditions in optimal control theory, see Michel (1982) and Seierstad and Sydsaeter (1987).
As a point of departure, consider first the problem of measuring welfare change along the first-best optimal path that obeys equations (6a)-(6c), where the externalities associated with relative consumption concerns are fully internalized. This constitutes a natural reference case, although it is presumably not very realistic. We use the superscript * to denote the socially optimal resource allocation, such that
{
c l kt*, ,t* t*,ltp,*}
∀ tsatisfy equations (4) and (6) along with the initial and terminal conditions for the capital stock, and then define the corresponding optimal value function at time t as follows:
* * * * ( )
( , , ) s t
t s s s
t
V =∞
∫
u c z c e−θ − ds. (8) The welfare change over the short time interval ( ,t t+dt) is given by the time derivative of equation (8), i.e.,*
* * * * *
( , , )
t
t t t t t
dV V V c z c
dt ≡ =θ −u . (9)
Defining genuine saving at any time t as ltk , where t l lt = tpeθt denotes the current value shadow price of physical capital, our first result is summarized as follows:
Observation 1. In a first-best optimum, genuine saving constitutes an exact measure of welfare change such that
* * *
t t t
V =l k . (10)
Observation 1 is a standard result, which reproduces the welfare change-equivalence property of genuine saving in the context of the benchmark model. The left-hand side of equation (10) is the welfare change over the short time interval ( ,t t+dt), while the right-hand side is interpretable as the genuine saving for the model set out above measured in units of utility at the first-best social optimum. While our model for simplicity only contains a one-dimensional capital concept (or state variable), the physical capital stock, a generalization to several capital stocks is straightforward: the right-hand side of equation (10) would then simply be the sum of changes in the value of all relevant capital stocks (see also Section 5 below).
3.2 Unregulated Economy
Here we analyze the probably more realistic case where the externalities associated with relative consumption concerns are not internalized, implying that the first-order condition for private consumption is given by equation (7) instead of equation (6a), and that equation (10) is no longer valid. Let
{
c kt0, t0,ltp,0}
∀tdenote the resource allocation in the unregulated economy, and let the corresponding value function at time t be given by
0 0 0 0 ( )
( , , ) s t
t s s s
t
V =∞
∫
u c z c e−θ − ds. (11) Also, let rt = f l kk( , )t t denote the interest rate at time t, and s t s 0R− =
∫
t r dt t denote the sum of interest rates from t to s (where s > ). Now, by using t0 0 0 0 0
( , , )
t t t t t
V =θV −u c z c , (12)
we can derive the following result:
Proposition 1. In an unregulated economy with externalities caused by relative consumption concerns, the measure of welfare change takes the form
0 0 0 0 0
exp( )
t t t s s t s
t
V l k ∞α R− c ds
= − −
∫
(13)
where
0 0 0
0
0 0 0 0 0 0
( , , )
( , , ) ( , , )
t t t
t
c t t t t t t
u c z
u c z u c z
α ∆
∆
= ∆
∆ + ∆ .
Proof: See Appendix.
The right-hand side of equation (13) is written as the product of the real welfare change (the expression within square brackets) and the marginal utility of consumption. Proposition 1 implies that the conventional measure of genuine saving, ltk , does not generally constitute t an exact measure of welfare change in an unregulated economy.9 The second term on the
9 The insight that externalities may change the principles of measuring welfare and welfare change is, of course, not new. Earlier research on green national accounting shows that technological change and environmental externalities add additional components to measures of welfare and welfare change in unregulated economies (see Aronsson, Löfgren and Backlund, 2004, and Aronsson and Löfgren, 2010, as well as references therein).
The novelties of the present paper are that it (i) shows how conspicuous consumption modifies the principles of
right-hand side is the value of the marginal externality and is given by a weighted sum of discounted future changes in the reference consumption, where the instantaneous weights are defined by the degrees of positionality along the economy’s equilibrium path. This component is forward looking, since the welfare function at any time t is intertemporal and reflects future utility.
Note that in the economy with identical individuals analyzed so far, c is always equal to c, implying that their growth rates will also be the same. Thus, in a growing economy, the second term on the right-hand side of equation (13) will be negative and genuine saving will overestimate the welfare change. Also, the more positional the consumers are, i.e., the larger the α, the greater the discrepancy between the conventional measure of genuine saving and the welfare change, ceteris paribus. Therefore, empirical estimates of the degree of positionality, along with estimates of changes in the average consumption, are important for calculating the second term on the right-hand side of equation (13).
What is the remaining informational content of ltk ? If we erroneously were to apply the t genuine saving formula given by equation (10) in an unregulated economy, it follows that
tkt
l does not constitute an indicator of local sustainable development as it does not measure the change in welfare over the time interval ( ,t t+dt). In other words, l >tkt 0 does not indicate a welfare improvement. Similarly, ltkt <0 does not in general imply a welfare decline. However, if consumption increases along the equilibrium path, ltk can still be used t for a one-sided test of local unsustainable development, since ltkt <0 then implies that the right-hand side of equation (13) is negative.
4 Adding a Catching-up-with-the-Joneses Mechanism
So far, we have assumed that the relative consumption comparisons are atemporal: each individual compares his/her current consumption with other people’s current consumption.
Although a great deal of empirical evidence on relative consumption concerns is interpretable as mainly reflecting such comparisons, it is easy to argue that intertemporal consumption comparisons may also be relevant to consider. For example, Senik (2010) presents empirical evidence consistent with such catching-up-with-the-Joneses comparisons by showing that measuring welfare-equivalent saving (an issue that to our knowledge has never been addressed) and (ii) by relating the welfare change measure to empirical measures of positional concerns.
individual well-being is higher if the individual’s standard of living exceeds that of his/her parents 15 years earlier, ceteris paribus. Such comparisons are also consistent with the empirical pattern of various financial puzzles (Constantinides, 1990; Campbell and Cochrane, 1999; and Díaz et al., 2003). It is therefore clearly interesting to investigate whether the qualitative results and interpretations presented in the previous section above carry over to a framework where the individual also derives utility from comparisons with other people’s past consumption. As will be shown, they essentially do.
There are at least two ways of modeling such intertemporal consumption comparisons.
One is to add a stock reflecting a weighted average of other people’s past consumption over the entire history of the model, as Aronsson and Löfgren (2008) did in their study on the comprehensive net national product under external habit formation. Another way is to assume that the individual directly derives utility from comparing his/her time t consumption with others’ consumption at time t− (and possibly other points in time as well). We use the latter ε approach as it requires less modifications of the model set out above. The instantaneous utility at time t is then rewritten to read
( , , , ) ( , , , )
t t t t t t t t t
U =u c z ∆ δ =u c z c c−ε , (14) where δt = −ct ct−ε denotes the individual’s relative consumption compared with others’ past consumption. Thus, the instantaneous utility is no longer time separable. All other aspects of the model remain as above.
We can then distinguish between the degree of atemporal positionality (which is analogous to the positionality concept discussed above) and the degree of intertemporal positionality.10 By using equation (14), these two measures are summarized as
( , , , )
(0,1) ( , , , ) ( , , , ) ( , , , )
t t t t
t
c t t t t t t t t t t t t
u c z
u c z u c z u c zδ
α δ
δ ∆∆ δ δ
= ∆ ∈
∆ + ∆ + ∆ (15)
( , , , )
(0,1) ( , , , ) ( , , , ) ( , , , )
t t t t
t
c t t t t t t t t t t t t
u c z
u c z u c z u c z
δ
δ
β δ
δ ∆ δ δ
= ∆ ∈
∆ + ∆ + ∆ . (16)
As before, α denotes the degree of atemporal positionality (with the same interpretation as t equation (2) above) and β the degree of intertemporal positionality at time t. The latter t positionality concept measures the fraction of the utility gain of an additional dollar spent on
10 This distinction originates from Aronsson and Johansson-Stenman (2014), who analyze the simultaneous implications of keeping-up-with-the-Joneses and catching-up-with-the-Joneses types of comparisons for optimal taxation in an OLG model.
consumption that is due to increased relative consumption compared with other people’s past consumption.
Consider first the socially optimal resource allocation. Here, since the reference consumption measure on which the intertemporal consumption comparison is based enters the instantaneous utility function with a time lag, equation (6a) is replaced with
( )
[ ( , , , ) ( , , , )] ( , , , ) 0
t t t
t t p
c c z c ct t t t ε c c z c ct t t t ε e θ c ct ε zt ε ct ε c et θ ε t
u − +u − − +u + + + + −l = , (17)
where the third term on the right-hand side is due to the delayed response mechanism caused by intertemporal consumption comparisons. The other first-order conditions remain as in equations (6b) and (6c). By using the same procedure as above, it is straightforward to show that equation (10) in Observation 1 still applies, since there are no uninternalized externalities that affect the welfare change measure, i.e., Vt* =lt*kt*.
In the unregulated economy, the first-order conditions are also in this case given by equations (6b), (6c), and (7), and we have the following analogue to Proposition 1:
Proposition 2. In an unregulated economy, where the externalities associated with relative consumption concerns are driven by both keeping-up-with-the-Joneses and catching-up-with- the-Joneses comparisons, the measure of welfare change takes the form
0 0 0 0 0 0 0
exp( ) exp( )
t t t s s t s s s t s
t t
V l k ∞α R− c ds ∞β +ε R+ −ε c ds
= − − − −
∫ ∫
. (18)
The interpretation of equation (18) is analogous to that of equation (13), with the only modification that the value of the marginal externality is divided in two parts in equation (18).
The reason is, of course, that equation (18) is based on the assumption that there are two mechanisms (a keeping-up and a catching-up mechanism) behind the relative consumption comparisons. We can see that externalities associated with the keeping-up and catching-up comparisons affect the welfare change measure in the same general way. Therefore, the interpretations of Proposition 1 essentially carry over to Proposition 2. While this conclusion is interesting per se, it also suggests that we may skip the catching-up component in what follows in order to keep the model as simple and transparent as possible.
5. Public Investments
This section extends the benchmark model by analyzing the role of public investments.
The implications of relative consumption concerns for the optimal provision of public goods have been addressed in several studies (e.g., Ng, 1987; Aronsson and Johansson-Stenman, 2008, forthcoming). Relative concerns for private consumption affect the optimal policy rule for public good provision via two channels: (1) an incentive to internalize positional externalities through increased public provision (which reduces the private consumption) and (2) an (indirect) incentive to reduce the public provision as relative consumption concerns lower the consumers’ marginal willingness to pay for public goods, ceteris paribus. As such, whether or not positional preferences for private consumption influence the policy rule for a public good largely depends on the preference elicitation format. In turn, this has a direct bearing on the way public investment ought to be reflected in the context of genuine saving, which motivates the following extension.
We will, consequently, add a public good of stock character to the instantaneous utility, which means that equation (1) changes to
( , , , ) ( , , , )
t t t t t t t t t
U =u c z g ∆ =u c z g c , (19)
where g denotes the level of the public good at time t (e.g., the state of the natural t environment). The public good accumulates through the following differential equation:11
t t t
g =q −gg , (20)
where q denotes the flow expenditure directed toward the public good at time t, i.e., the t instantaneous contribution, and g denotes the rate of depreciation. We also impose the initial condition that g is fixed and the terminal condition lim0 t→∞ gt ≥ . Finally, since part of 0 output is used for contributions to the public good, the resource constraint slightly changes such that
( , )
t t t t t
k = f l k − −c ρq , (21)
where ρ is interpretable as the (fixed) marginal rate of transformation between the public good and the private consumption good.
11 Since the public good is interpretable in terms of environmental quality, a possible (and realistic) extension of equation (20) would be to assume that increased output leads to lower environmental quality (instead of just assuming a natural rate of depreciation). Yet, we refrain from this extension here as it is not important for the qualitative results derived below.
These extensions mean that we have added the control variable q and the state variable g to the benchmark model; otherwise, the model is the same as in Section 2. Thus, the social decision problem is to choose c , t l , and t q for all t to maximize the present value of future t utility,
0 0
( , , , )
t t
t t t t t
U e θ dt u c z g c e θ dt
∞ ∞
− = −
∫ ∫
,subject to equations (20) and (21) along with initial and terminal conditions. The present value Hamiltonian can then be written as
( , , , ) [ ( , ) ] [ ]
p t p p
t t t t t t t t t t t t t
H =u c z g c e−θ +l f l k −ρq −c +µ q −gg . (22) The variable µ denotes the present value shadow price of the public good at time t, and its tp current value counterpart is given by µt =µtpeθt. Note that equations (6a)-(6e), if written in terms of the instantaneous utility function examined here (i.e., equation [19]), are necessary conditions also in this extended model. In addition to (the modified) equations (6), the first- order conditions for a social optimum also include an efficiency condition for q, an equation of motion for µ , and an additional transversality condition, i.e., p
p p
t t
l ρ µ= (23a)
( , , , )
p
p t t p
t g t t t t t
t
H c z g c e
g
µ ∂ u −θ µ g
= − = − +
∂ (23b)
limt→∞µtp ≥0 (=0 if limt→∞ gt > ). 0 (23c) Equations (6a)-(6e) and (23) characterize the social optimum. As in Subsections 3.1 and 3.2, if the positional externality has not become internalized, equation (6a) should be replaced with equation (7) (again modified to reflect the instantaneous utility function [19] that contains the public good). For further use, let us also solve the differential equation (23b) forward subject to the transversality condition (23c),12 which gives
(s t)
( , , , )
p s
t g s s s s
t
c z g c e θ e g ds
µ =∞
∫
u − − − . (24)We are now ready to present the main results. As before, to separate the social optimum from an allocation with uninternalized positional externalities, we use superscript * to denote the socially optimal resource allocation and superscript 0 to denote the unregulated economy.
12 We assume that limt→∞ gt >0.
The value function (social welfare function) at any time t will be defined in the same way as above, i.e.,
( )
( , , , ) s t
t s s s s
t
V =∞
∫
u c z g c e−θ − ds.Then, by recognizing that genuine saving now reads ltkt +µtg (since the capital concept is t two-dimensional here), we can use the same procedure as in Observation 1 and Proposition 1 to derive an analogue to equation (10) as follows:
* * * *
t t t t
V =l k +ρg , (25a)
and a corresponding analogue to equation (13):
0 0 0 0 0 0
exp( )
t t t t s s t s
t
V l k ρg ∞α R− c ds
= + − −
∫
. (25b)
Except for the public investment component, equations (25a) and (25b) are interpretable in exactly the same way as their counterparts in the simpler benchmark model, i.e., equations (10) and (13).13
Equations (25a) and (25b) together imply the following treatment of public investment:
Proposition 3. Irrespective of whether the resource allocation is first best or represented by an unregulated equilibrium, the accounting price of public investment is given by the marginal rate of transformation between the public good and the private consumption good,
ρ .
Proposition 3 has a strong implication, as it means that the valuation of public investment might be based on observables and that the same valuation procedure applies in a social optimum and a distorted market economy. Although convenient, this result may seem surprising at first sight. Yet, note that the information content in ρ differs between the two regimes due to differences in the underlying policy rules for public provision. To see this more clearly, let
13 If we extend the model by allowing ρ to vary over time, e.g., due to disembodied technological change, an additional (forward-looking) term representing the marginal value of this technological change must be added to equations (25a) and (25b).
g g gc
c c
MRS u
u u
u
u ∆
= =
+
denote the marginal rate of substitution between the public good and private consumption measured with the reference consumption, c , held constant, and
g g
gc
c c c
CMRS u
u u
=u u = +
denote the corresponding marginal rate of substitution measured with the relative consumption, ∆, held constant. Therefore, MRSgcrefers to a conventional marginal willingness to pay measure with other people’s consumption held constant. This means that an increase in the public good will not only imply reduced (absolute) consumption, the individual will also take into account the fact that his/her relative consumption decreases.
CMRSgc, on the other hand, reflects each respondent’s marginal willingness to pay for the public good conditional on the fact that other people have to pay the same amount, implying that relative consumption is held fixed. Thus, in this case there is no additional cost in terms of reduced relative consumption, implying that under relative consumption concerns
CMRS will exceed gc MRS . Indeed, it is straightforward to show that gc / (1 )
gc gc
CMRS =MRS −α .
By using equations (6a), (23a), and (24), the policy rule for public provision in the first- best optimum is then given by
* *
( ( )) , ( ( ))
*
* , *
1
s t s gc s t
R s t R s t
t
s gc
t t t s
CMRS e g ds MRS e g ds
µ ρ
l − α −
∞ ∞
− + − − + −
= = =
∫ ∫
− , (26a)while the corresponding policy rule implicit in the unregulated economy, based on equations (7), (23a), and (24), becomes
0
( ( ))
0 0 ,
Rs t s t t
s gc
t t
MRS e g ds
µ ρ
l −
∞ − + −
=
∫
= . (26b)Equations (26a) and (26b) are different variants of the Samuelson condition for a state variable public good. In a social optimum, where all positional externalities are internalized, the social marginal rate of substitution between the public good and private consumption is given by CMRSgc, which recognizes that relative consumption is pure waste. In the unregulated economy, on the other hand, agents behave as if others’ consumption, c , is exogenous, meaning that the marginal rate of substitution implicit in the first-order conditions
is given by MRSgc. In either case, however, it is optimal to equate the weighted sum of marginal rates of substitution with the marginal rate of transformation, which explains Proposition 3. There is an important practical implication here: regardless of whether we are in a first-best economy or in an unregulated equilibrium, public investment can be valued by the instantaneous marginal cost, such that there is no need to estimate future generations’
marginal willingness to pay for current additions to the public good.
6. Heterogeneity, Redistribution Policy and Genuine Saving
In the preceding sections we have consistently considered a representative-agent economy, which is the typical framework used in earlier literature dealing with genuine saving. In this section, we extend the analysis to a model where consumers are heterogeneous in terms of productivity and productivity is private information. Such an extension is of course not without costs in terms of greater complexity and less transparency. However, the benefits are also considerable in terms of greater realism. In particular, it acknowledges the fact that there is a social value of increased equality, and it also takes into account that redistributive taxes are generally distortionary in the sense that they affect people’s incentive to work. This framework, which is common in earlier literature on optimal taxation, has recently been used to analyze the policy implications of relative consumption concerns. As explained in the introduction, such an extension enables us to generalize the study of genuine saving to a second-best economy where the social planner redistributes and internalizes externalities subject to an incentive constraint.14
We make two simplifying assumptions. First, we do not consider public investments. Since Proposition 3 can be shown to apply also in the model set out below, little additional insight would be gained from studying such investments here as well. Second, to avoid unnecessary technical complications with many different consumer types, we use the two-type setting originally developed by Stern (1982) and Stiglitz (1982). The consumers differ in productivity, and the high type (type 2) is more productive (measured in terms of the before-
14 To our knowledge, the paper by Aronsson, Cialani, and Löfgren (2012) is the only earlier study dealing with genuine saving in a second-best economy with distortionary taxation. Their study is based on a representative- agent model (and as such does not consider redistribution policy) and assumes that individuals only care about their own absolute consumption and use of leisure (meaning that relative consumption is not dealt with at all).
tax wage rate) than the low type (type 1).15 The population is constant and normalized to one for notational convenience, and there is a constant share, ni, of individuals of type i, such that
i 1
in =
∑
.6.1 Preferences and Social Decision-Problem
We also allow for type-specific differences in preferences. The instantaneous utility function facing each individual of type i can then be written as
( , , ) ( , , )
i i i i i i i i
t t t t t t t
U =u c z ∆ =u c z c , (27) where ∆ = − denotes the relative consumption of type i. The functions ( )it cit ct ui ⋅ and ( )u ⋅ in i equation (27) have the same general properties as their counterparts in equation (1). Also, and similarly to the benchmark model presented in Section 2, the relative consumption concerns in equation (27) solely reflect comparisons with other people’s current consumption, i.e., we abstract from the catching-up mechanism briefly discussed above. The reference level is assumed to be the average consumption in the economy as a whole, ct =n c1 1t +n c2 t2.16 As before, each consumer is small relative to the overall economy and hence treats c as t exogenous.
In a way similar to Sections 2 and 3, it is useful to be able to measure the extent to which relative consumption matters for individual utility. By using the function ui( )⋅ in equation (27), we can define the degree of positionality of type i at time t such that
( , , )
(0,1) ( , , ) ( , , )
i i i i
i t t t
t i i i i i i i i
c t t t t t t
u c z
u c z u c z
α ∆
∆
= ∆ ∈
∆ + ∆ for i=1,2, all t. (28) Therefore, α measures the fraction of the utility gain for type i at time t of increased ti consumption that is due to increased relative consumption (compared with the reference consumption level).
15 Aronsson (2010) uses a similar two-type model to derive a second-best analogue to the comprehensive net national product. However, his study neither addresses the implications of relative consumption concerns nor examines genuine saving, which are the main issues here.
16 This is the most common assumption in the literature dealing with optimal policy responses to relative consumption concerns. Although the definition of reference consumption at the individual level (i.e., whether it is based on the economy-wide mean value or reflects more narrow social reference groups) matters for public policy, it is not equally important here, where the main purpose is to characterize an aggregate measure of social savings.
Since we assume that all individuals compare their own consumption with the average consumption in the economy as a whole, we can define the positional externality in terms of the average degree of positionality. The average degree of positionality measured over all individuals at time t can be written as (recall that the total population size is normalized to unity)
1 1 2 2
(0,1)
t n t n t
α = α + α ∈ , (29)
which also reflects the sum of the marginal willingness to pay to avoid the positional consumption externality measured per unit of consumption. Estimates of α can be found in t empirical literature on relative consumption concerns, as discussed above (see the introduction).
The social objective is assumed to be a general social welfare function
(
1 2)
0 0
, t
t t
W =∞
∫
ω U U e−θ dt. (30)The instantaneous social welfare function, ω( )⋅ , is differentiable and increasing in the instantaneous individual utilities. The resource constraint can now be written as
1 2
( , , ) i i
t t t t i t
k = f k −
∑
n c , (31)where it =n li it is interpretable as the aggregate input of type i labor at time t. As before, the production function, f( , 1t 2t,kt), is characterized by constant returns to scale.
Following Aronsson and Johansson-Stenman (2010), we assume that the social planner (or government) can observe labor income and saving at the individual level, while ability is private information. We also assume that the social planner wants to redistribute from the high type to the low type. To eliminate the incentive for the high type to mimic the low type (in order to gain from this redistribution profile), we impose a self-selection constraint such that each individual of the high type weakly prefers the allocation intended for his/her type over the allocation intended for the low type. Also, to simplify the analysis, we assume that the social planner commits to the resource allocation decided at time zero.17 One way to rationalize this assumption in our framework is to interpret the model in terms of a continuum of perfectly altruistic generations, i.e., dynasties (instead of single consumers with infinite
17 See Brett and Weymark (2008) for an analysis of optimal taxation without commitment based on the self- selection approach to optimal taxation.
